Question

Derive an expression for the velocity of transverse waves in a stretched uniform string.

Answer 1

Answer:

Explanation:

=> We can generate a transverse wave on a stretched uniform string.

=> The speed of transverse wave set up in the string depends upon few factors which includes tension applied (T), mass per unit length (μ) of the string etc.

=> We can derive an expression for the velocity (V) of transverse waves in a stretched uniform string with the help of below method:

For the element, Force$F_{element} = d_{map}$ = μ$dx*d^2y/dt^2$

Tsinθ2 - Tsinθ1 = μdx*$d^2y/dt^2$

T[tanθ2 - tanθ1] = μdx*$d^2y/dt^2$

T[$\frac{dy}{dx}|_{x2}  -\frac{dy}{dx}|_{x1}$] = μdx*$d^2y/dt^2$

T[$\frac{d}{dx} [\frac{dy}{dx}]dx$] =  μdx*$d^2y/dt^2$

$\frac{d^2y}{dx^2} = \frac{1}{T/u} \frac{d^2y}{dt^2}$

v^2 = T/μ

=> v = √T/ μ